Axion-like fields are naturally generated by a mechanism of anomaly cancellation of one or more anomalous gauge abelian symmetries at the Planck scale , emerging as duals of a two-form from the massless bosonic sector of string theory .
This suggests an analogy of the Green-Schwarz mechanism of anomaly cancellation , at field theory level , which results in one or more Stueckelberg pseudoscalars .
In the case of a single Stueckelberg pseudoscalar b , vacuum misalignments at phase transitions in the early Universe at the GUT scale provide a small mass - due to instanton suppression of the periodic potential - for a component of b , denoted as \chi and termed the ” axi-Higgs ” , which is a physical axion-like particle .
The coupling of the axi-Higgs to the gauge sector via Wess-Zumino terms is suppressed by the Planck mass , which guarantees its decoupling , while its angle of misalignment is related to M _ { GUT } .
We build a gauged E _ { 6 } \times U ( 1 ) model with anomalous U ( 1 ) .
It contains both an automatic invisible QCD axion and an ultra-light axi-Higgs .
The invisible axion present in the model solves the strong CP problem and has mass in the conventional range while the axi-Higgs , which can act as dark matter , is sufficiently light ( 10 ^ { -22 } \textrm { eV } < m _ { \chi } < 10 ^ { -20 } \textrm { eV } ) to solve short-distance problems which confront other cold dark matter candidates .